In this paper we study various forms of (hereditary) structural completeness for quasivarieties of algebras, using mostly algebraic techniques. More specifically we study relative weakly projective algebras and the way they interact with structural completeness in quasivarieties. These ideas are then applied to the study of $C$-structural completeness and $C$-primitivity, through an algebraic generalization of Prucnal's substitution. Finally we study in depth dual i-discriminator quasivarieties in which a particular instance of Prucnal's substitution is used to prove that if each fundamental operation commutes with the i-discriminator, then it is primitive.